Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$.
Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$.
Can we get an upper bound for $\pi_f(x)$?
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$\begingroup$ Well, it is an open question whether $\pi_f(x)$ is at most a constant for all $f$ and $x$. See en.wikipedia.org/wiki/Bunyakovsky_conjecture $\endgroup$– Tony HuynhCommented Jul 16, 2016 at 21:19
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1 Answer
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Yes, you can get sharp upper bound, matching (up to a multiplicative constant) the asymptotic predicted by the Bateman-Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture). This is a standard (and almost the first) application of sieve methods. I recommend the books by Halberstam and Richert, or Cojocaru and Murty.
